\( a \frac{18^{a} \cdot{ }_{15} a+1}{30^{a-1} 3 a} \times \frac{\left(3^{a} b\right)^{2}}{q^{a-1}} \div \frac{3 b^{a+1}}{18^{a-1}} \) b. \( \frac{\left(a b^{2} c\right)^{2}}{a^{-1} b^{4} c^{3}} \times \frac{\left(a^{-2} b^{3} c^{5}\right)^{0}}{a^{-1} b^{4} c^{2}} \) c. \( \frac{x^{a} y^{2 a}}{x^{2 a^{-1}}} \div \frac{\left(x^{2} y\right)^{a}}{x^{3+a} y^{-1}} \) e \( \frac{\left(a b^{2}\right)^{-6} \times\left(a^{3} b^{4}\right)^{2}}{\left(a^{0} b^{2}\right)^{-\frac{1}{2}}} \) \( 3^{a-1} \cdot b^{(a+2)} \cdot 8^{2} \) \( 1 z^{a}-13^{a} \) \( \left(x y z^{2}\right)^{-1} \div\left(x^{3} y^{2} z^{0}\right)^{-1} \times \frac{x y^{2} z^{4}}{x^{-3} y^{2} z^{0}} \)

Simplify the expression by following steps: - step0: Simplify: \(3^{a-1}b^{\left(a+2\right)}\times 8^{2}\) - step1: Remove the parentheses: \(3^{a-1}b^{a+2}\times 8^{2}\) - step2: Rewrite the expression: \(3^{a-1}b^{a+2}\times 64\) - step3: Reorder the terms: \(64\times 3^{a-1}b^{a+2}\) Expand the expression \( \left(x y z^{2}\right)^{-1} \div\left(x^{3} y^{2} z^{0}\right)^{-1} \times \frac{x y^{2} z^{4}}{x^{-3} y^{2} z^{0}} \) Simplify the expression by following steps: - step0: Simplify: \(\left(xyz^{2}\right)^{-1}\div \left(x^{3}y^{2}z^{0}\right)^{-1}\times \frac{xy^{2}z^{4}}{x^{-3}y^{2}z^{0}}\) - step1: Evaluate the power: \(\left(xyz^{2}\right)^{-1}\div \left(x^{3}y^{2}\times 1\right)^{-1}\times \frac{xy^{2}z^{4}}{x^{-3}y^{2}z^{0}}\) - step2: Multiply the terms: \(\left(xyz^{2}\right)^{-1}\div \left(x^{3}y^{2}\right)^{-1}\times \frac{xy^{2}z^{4}}{x^{-3}y^{2}z^{0}}\) - step3: Evaluate the power: \(\left(xyz^{2}\right)^{-1}\div \left(x^{3}y^{2}\right)^{-1}\times \frac{xy^{2}z^{4}}{x^{-3}y^{2}\times 1}\) - step4: Multiply the terms: \(\left(xyz^{2}\right)^{-1}\div \left(x^{3}y^{2}\right)^{-1}\times \frac{xy^{2}z^{4}}{x^{-3}y^{2}}\) - step5: Divide the terms: \(\left(xyz^{2}\right)^{-1}\div \left(x^{3}y^{2}\right)^{-1}x^{4}z^{4}\) - step6: Divide the terms: \(\frac{x^{2}y}{z^{2}}\times x^{4}z^{4}\) - step7: Reduce the fraction: \(x^{2}yx^{4}z^{2}\) - step8: Multiply the terms: \(x^{6}yz^{2}\) Expand the expression \( 1 z^{a}-13^{a} \) Simplify the expression by following steps: - step0: Multiply: \(1\times z^{a}-13^{a}\) - step1: Multiply: \(z^{a}-13^{a}\) Expand the expression \( \frac{x^{a} y^{2 a}}{x^{2 a^{-1}}} \div \frac{\left(x^{2} y\right)^{a}}{x^{3+a} y^{-1}} \) Simplify the expression by following steps: - step0: Calculate: \(\frac{x^{a}y^{2a}}{x^{2a^{-1}}}\div \frac{\left(x^{2}y\right)^{a}}{x^{3+a}y^{-1}}\) - step1: Multiply the terms: \(\frac{\left(xy^{2}\right)^{a}}{x^{2a^{-1}}}\div \frac{\left(x^{2}y\right)^{a}}{x^{3+a}y^{-1}}\) - step2: Multiply the terms: \(\frac{\left(xy^{2}\right)^{a}}{x^{2a^{-1}}}\div \frac{\left(x^{2}y\right)^{a}}{y^{-1}x^{3+a}}\) - step3: Divide the terms: \(x^{a-2a^{-1}}y^{2a}\div \frac{\left(x^{2}y\right)^{a}}{y^{-1}x^{3+a}}\) - step4: Divide the terms: \(x^{a-2a^{-1}}y^{2a}\div \left(y^{a+1}x^{a-3}\right)\) - step5: Rewrite the expression: \(\frac{x^{a-2a^{-1}}y^{2a}}{y^{a+1}x^{a-3}}\) - step6: Reduce the fraction: \(\frac{y^{a-1}}{x^{-3+2a^{-1}}}\) - step7: Simplify: \(\frac{y^{a-1}}{x^{-3+\frac{2}{a}}}\) - step8: Add the terms: \(\frac{y^{a-1}}{x^{\frac{-3a+2}{a}}}\) Expand the expression \( \frac{\left(a b^{2}\right)^{-6} \times\left(a^{3} b^{4}\right)^{2}}{\left(a^{0} b^{2}\right)^{-\frac{1}{2}} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{\left(ab^{2}\right)^{-6}\left(a^{3}b^{4}\right)^{2}}{\left(a^{0}b^{2}\right)^{-\frac{1}{2}}}\) - step1: Evaluate the power: \(\frac{\left(ab^{2}\right)^{-6}\left(a^{3}b^{4}\right)^{2}}{\left(1\times b^{2}\right)^{-\frac{1}{2}}}\) - step2: Multiply: \(\frac{\left(ab^{2}\right)^{-6}\left(a^{3}b^{4}\right)^{2}}{\left(b^{2}\right)^{-\frac{1}{2}}}\) - step3: Evaluate the power: \(\frac{\left(ab^{2}\right)^{-6}\left(a^{3}b^{4}\right)^{2}}{b^{-1}}\) - step4: Multiply the terms: \(\frac{\frac{1}{b^{4}}}{b^{-1}}\) - step5: Express with a positive exponent: \(\frac{\frac{1}{b^{4}}}{\frac{1}{b}}\) - step6: Multiply by the reciprocal: \(\frac{1}{b^{4}}\times b\) - step7: Reduce the fraction: \(\frac{1}{b^{3}}\times 1\) - step8: Multiply the terms: \(\frac{1}{b^{3}}\) Expand the expression \( \frac{\left(a b^{2} c\right)^{2}}{a^{-1} b^{4} c^{3}} \times \frac{\left(a^{-2} b^{3} c^{5}\right)^{0}}{a^{-1} b^{4} c^{2}} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{\left(ab^{2}c\right)^{2}}{a^{-1}b^{4}c^{3}}\times \frac{\left(a^{-2}b^{3}c^{5}\right)^{0}}{a^{-1}b^{4}c^{2}}\) - step1: Divide the terms: \(\frac{a^{3}}{c}\times \frac{\left(a^{-2}b^{3}c^{5}\right)^{0}}{a^{-1}b^{4}c^{2}}\) - step2: Divide the terms: \(\frac{a^{3}}{c}\times \frac{a}{b^{4}c^{2}}\) - step3: Multiply the terms: \(\frac{a^{3}\times a}{cb^{4}c^{2}}\) - step4: Multiply the terms: \(\frac{a^{4}}{cb^{4}c^{2}}\) - step5: Multiply the terms: \(\frac{a^{4}}{c^{3}b^{4}}\) Expand the expression \( a \frac{18^{a} \cdot{ }_{15} a+1}{30^{a-1} 3 a} \times \frac{\left(3^{a} b\right)^{2}}{q^{a-1}} \div \frac{3 b^{a+1}}{18^{a-1}} \) Simplify the expression by following steps: - step0: Calculate: \(a\times \frac{18^{a}a+1}{30^{a-1}\times 3a}\times \frac{\left(3^{a}b\right)^{2}}{q^{a-1}}\div \frac{3b^{a+1}}{18^{a-1}}\) - step1: Multiply the terms: \(a\times \frac{18^{a}a+1}{3\times 30^{a-1}a}\times \frac{\left(3^{a}b\right)^{2}}{q^{a-1}}\div \frac{3b^{a+1}}{18^{a-1}}\) - step2: Divide the terms: \(a\times \frac{18^{a}a+1}{3\times 30^{a-1}a}\times \frac{\left(3^{a}b\right)^{2}}{q^{a-1}}\div \frac{3^{3-2a}b^{a+1}}{2^{a-1}}\) - step3: Multiply the terms: \(\frac{18^{a}a\left(3^{a-1}b\right)^{2}+\left(3^{a-1}b\right)^{2}}{10\times 30^{a-2}q^{a-1}}\div \frac{3^{3-2a}b^{a+1}}{2^{a-1}}\) - step4: Multiply by the reciprocal: \(\frac{18^{a}a\left(3^{a-1}b\right)^{2}+\left(3^{a-1}b\right)^{2}}{10\times 30^{a-2}q^{a-1}}\times \frac{2^{a-1}}{3^{3-2a}b^{a+1}}\) - step5: Rewrite the expression: \(\frac{b^{2}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{10\times 30^{a-2}q^{a-1}}\times \frac{2^{a-1}}{3^{3-2a}b^{a+1}}\) - step6: Reduce the fraction: \(\frac{b^{1-a}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{10\times 30^{a-2}q^{a-1}}\times \frac{2^{a-1}}{3^{3-2a}}\) - step7: Rewrite the expression: \(\frac{b^{1-a}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{10\times 2^{a-2}\times 15^{a-2}q^{a-1}}\times \frac{2^{a-1}}{3^{3-2a}}\) - step8: Reduce the fraction: \(\frac{b^{1-a}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{10\times 15^{a-2}q^{a-1}}\times \frac{2}{3^{3-2a}}\) - step9: Multiply the terms: \(\frac{b^{1-a}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)\times 2}{10\times 15^{a-2}q^{a-1}\times 3^{3-2a}}\) - step10: Multiply the terms: \(\frac{b^{1-a}\left(2a\left(3^{a-1}\right)^{2}\times 18^{a}+2\left(3^{a-1}\right)^{2}\right)}{10\times 15^{a-2}q^{a-1}\times 3^{3-2a}}\) - step11: Multiply the terms: \(\frac{b^{1-a}\left(2a\left(3^{a-1}\right)^{2}\times 18^{a}+2\left(3^{a-1}\right)^{2}\right)}{10\times 5^{a-2}\times 3^{-a+1}q^{a-1}}\) - step12: Factor: \(\frac{b^{1-a}\times 2\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{10\times 5^{a-2}\times 3^{-a+1}q^{a-1}}\) - step13: Reduce the fraction: \(\frac{b^{1-a}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{5\times 5^{a-2}\times 3^{-a+1}q^{a-1}}\) - step14: Simplify: \(\frac{b^{1-a}\left(a\left(3^{a-1}\right)^{2}\times 18^{a}+\left(3^{a-1}\right)^{2}\right)}{\left(5q\right)^{a-1}\times 3^{-a+1}}\) - step15: Simplify: \(\frac{b^{1-a}\left(a\times 18^{a}+1\right)}{\left(5q\right)^{a-1}\times 3^{-a+1}}\) - step16: Expand the expression: \(\frac{ab^{1-a}\times 18^{a}+b^{1-a}}{\left(5q\right)^{a-1}\times 3^{-a+1}}\) Let's break down each expression step by step: ### a. \[ a \frac{18^{a} \cdot{ }_{15} a+1}{30^{a-1} 3 a} \times \frac{\left(3^{a} b\right)^{2}}{q^{a-1}} \div \frac{3 b^{a+1}}{18^{a-1}} \] 1. **Simplifying the first part:** \[ \frac{18^{a} \cdot{ }_{15} a+1}{30^{a-1} 3 a} \] This can be simplified to: \[ \frac{ab^{1-a} \times 18^{a} + b^{1-a}}{(5q)^{a-1} \times 3^{-a+1}} \] 2. **Combining with the second part:** \[ \frac{\left(3^{a} b\right)^{2}}{q^{a-1}} = \frac{9^{a} b^{2}}{q^{a-1}} \] 3. **Dividing by the last part:** \[ \div \frac{3 b^{a+1}}{18^{a-1}} = \frac{3 b^{a+1}}{18^{a-1}} = \frac{3 b^{a+1}}{18^{a-1}} \] Final result: \[ \frac{ab^{1-a} \times 18^{a} + b^{1-a}}{(5q)^{a-1} \times 3^{-a+1}} \] ### b. \[ \frac{\left(a b^{2} c\right)^{2}}{a^{-1} b^{4} c^{3}} \times \frac{\left(a^{-2} b^{3} c^{5}\right)^{0}}{a^{-1} b^{4} c^{2}} \] 1. **Simplifying the first part:** \[ \left(a b^{2} c\right)^{2} = a^{2} b^{4} c^{2} \] Thus, \[ \frac{a^{2} b^{4} c^{2}}{a^{-1} b^{4} c^{3}} = a^{2+1} c^{2-3} = a^{3} c^{-1} \] 2. **The second part simplifies to:** \[ \left(a^{-2} b^{3} c^{5}\right)^{0} = 1 \] Final result: \[ \frac{a^{4}}{c^{3}b^{4}} \] ### c. \[ \frac{x^{a} y^{2 a}}{x^{2 a^{-1}}} \div \frac{\left(x^{2} y\right)^{a}}{x^{3+a} y^{-1}} \] 1. **Simplifying the first part:** \[ \frac{x^{a} y^{2 a}}{x^{2 a^{-1}}} = x^{a - 2 a^{-1}} y^{2 a} \] 2. **The second part simplifies to:** \[ \frac{\left(x^{2} y\right)^{a}}{x^{3+a} y^{-1}} = \frac{x^{2a} y^{a}}{x^{3+a} y^{-1}} = x^{2a - (3+a)} y^{a + 1} \] Final result: \[ \frac{y^{a-1}}{x^{\frac{-3a+2}{a}}} \] ### d. \[ \frac{\left(a b^{2}\right)^{-6} \times\left(a^{3} b^{4}\right)^{2}}{\left(a^{0} b^{2}\right)^{-\frac{1}{2}}} \] 1. **Simplifying the numerator:** \[ \left(a b^{2}\right)^{-6} = a^{-6} b^{-12} \] \[ \left(a^{3} b^{4}\right)^{2} = a^{6} b^{8} \] 2. **Combining:** \[ a^{-6} b^{-12} \times a^{6} b^{8} = b^{-4} \] 3. **The denominator simplifies to:** \[ \left(a^{0} b^{2}\right)^{-\frac{1}{2}} = b^{-1} \] Final result: \[ \frac{1}{b^{3}} \] ### e. \[ 3^{a-1} \cdot b^{(a+2)} \cdot 8^{2} \] 1. **Calculating:** \[ 8^{2} = 64 \] Final result: \[ 64 \times 3^{a-1} b^{a+2} \] ### f. \[ 1 z^{a}-13^{a} \] Final result: \[ z^{a} - 13^{a} \] ### g. \[ \left(x y z^{2}\right)^{-1} \div\left(x^{3} y^{2} z^{0}\right)^{-1} \times \frac{x y^{2} z^{4}}{x^{-3} y^{2} z^{0}} \] 1. **Simplifying the first part:** \[ \left(x y z^{2}\right)^{-1} = x^{-1} y^{-1} z^{-2} \] \[ \left(x^{3} y^{2} z^{0}\right)^{-1} = x^{-3} y^{-2} \] 2. **Dividing:** \[ x^{-1} y^{-1} z^{-2} \div x^{-3} y^{-2} = x^{2} y^{1} z^{-2} \] 3. **Combining with the last part:** \[ \frac{x y^{2} z^{4}}{x^{-3} y^{2} z^{0}} = x^{4} z^{4} \] Final result: \[ x^{6} y z^{2} \] ### Summary of Results: - a: \(\frac{ab^{1-a} \times 18^{a} + b^{1-a}}{(5q)^{a-1} \times 3^{-a+1}}\) - b: \(\frac{a^{4}}{c^{3}b^{4}}\) - c